It is known that all amenable groups do not contain free subgroups (of rank>1). But there are amenable groups containing free semigroups. Which amenable groups cannot contain free semigroups?

1$\begingroup$ Well, obviously a group containing a free semigroup has exponential growth. So a more specific question might be 'Is there an example of an amenable group with exponential growth but no free subsemigroup?'. $\endgroup$ – HJRW Feb 27 '11 at 1:41
This is the answer to the question asked by Henry. The wreath product $\mathbb Z_2 {\rm wr} G$, where $G$ is the Grigorchuk (torsion) group of subexponential growth, obviously has exponential growth and is amenable and torsion. In particular, it has no free subsemigroups.
For elementary amenable (in particular, solvable) groups, existence of noncyclic free subsemigroups is equivalent to exponential growth [C. Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), 3, 396407].

$\begingroup$ Thank you I was not aware of this result. $\endgroup$ – Mustafa Gokhan Benli Feb 28 '11 at 6:04